| 1. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Boltzmann equation and hydrodynamics at the Burnett level
- 2012
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences. - 1937-5093 .- 1937-5077. ; 5:2, s. 237-260
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Tidskriftsartikel (refereegranskat)abstract
- The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.
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| 2. |
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| 3. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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DSMC Methods for Multicomponent Plasmas
- 2012. - 1
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Ingår i: DSMC and Related Simulations. - New York : American Institute of Physics (AIP). - 9780735411159 ; , s. 541-548
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Konferensbidrag (refereegranskat)abstract
- A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of the Landau-Fokker-Planck equations by the Boltzmann equations of a quasi-Maxwellian kind. This means that the total collision frequency for the corresponding Boltzmann equation does not depend on velocities. This allows one to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes (as particular cases) the well-known methods of Takizuka & Abe(1977) and Nanbu(1997) and generalizes the approach of Bobylev & Nanbu(2000). The numerical scheme of this paper is simpler than the schemes by Takizuka & Abe and by Nanbu. We derive it for the general case of multicomponent plasmas
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| 4. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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DSMC Modeling of a Single Hot Spot Evolution Using the Landau-Fokker-Planck Equation
- 2014
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Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - : Springer Science and Business Media LLC. - 0167-8019 .- 1572-9036. ; 132:1, s. 107-116
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Tidskriftsartikel (refereegranskat)abstract
- Numerical solution of a fully nonlinear one dimensional in space and three dimensional in velocity space electron kinetic equation is presented. Direct Simulation Monte Carlo (DSMC) method used for the nonlinear Landau-Fokker-Planck (LFP) collision operator is combined with Particle-in-Cell (PiC) simulations. An assumption of a self-consistent ambipolar electric field is used. The illustrative simulation results for the relaxation of the initial temperature perturbation are compared with the antecedent analytical and numerical results.
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| 5. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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From Particle Systems to the Landau Equations : A Consistency Result
- 2013
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Ingår i: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 319:3, s. 693-702
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Tidskriftsartikel (refereegranskat)abstract
- We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey to the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in term of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.
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| 6. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Kinetic modeling of economic games with large number of participants
- 2011
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences. - 1937-5093 .- 1937-5077. ; 4:1, s. 169-185
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Tidskriftsartikel (refereegranskat)abstract
- We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters where is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.
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| 7. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Monte-Carlo method for two component plasmas
- 2012
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Ingår i: Matematicheskoe Modelirovanie. - Moskva : Steklov Mathematical Institute, Russian Academy of Sciences. - 0234-0879. ; 24:9, s. 35-49
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Tidskriftsartikel (refereegranskat)abstract
- The new direct simulation method of Monte-Carlo type (DSMC) for Coulomb collisions in the case of two component plasma is considered. A brief literature review and preliminary information concerning the problem are given. Further the idea that lies in the basis of the method is discussed and its scheme is provided. The illustrative numerical simulation of the initial distribution relaxation for one and two sorts of particles in 3D case in the velocity space is performed. Simulation results are compared with the numerical results based on the completely conservative finite difference schemes for the Landau-Fokker-Planck equation. Estimation of calculation accuracy obtained from numerical results is given.
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| 8. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas
- 2013
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 246, s. 123-144
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Tidskriftsartikel (refereegranskat)abstract
- A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck equations by Boltzmann equations of quasi-Maxwellian kind. It means that the total collision frequency for the corresponding Boltzmann equation does not depend on the velocities. This allows to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes the well-known methods of Takizuka and Abe (1977) [12] and Nanbu (1997) as particular cases, and generalizes the approach of Bobylev and Nanbu (2000). The numerical scheme of this paper is simpler than the schemes by Takizuka and Abe [12] and by Nanbu. We derive it for the general case of multicomponent plasmas and show some numerical tests for the two-component (electrons and ions) case. An optimal choice of parameters for speeding up the computations is also discussed. It is also proved that the order of approximation is not worse than O(root epsilon), where epsilon is a parameter of approximation being equivalent to the time step Delta t in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu. (C) 2013 Elsevier Inc. All rights reserved.
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| 9. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Shock wave structure for generalized Burnett equations
- 2011
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Ingår i: Physics of fluids. - New York : American Institute of Physics (AIP). - 1070-6631 .- 1089-7666. ; 23:3, s. 030607-030607-10
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Tidskriftsartikel (refereegranskat)abstract
- Stationary shock wave solutions for the generalized Burnett equations (GBE) [ A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys. 132, 569 (2008) ] are studied. Based on the results of Bisi et al. [Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models 1, 295 (2008) ], we choose a unique (optimal) form of GBE and solve numerically the shock wave problem for various Mach numbers. The results are compared with the numerical solutions of NavierStokes equations and with the MottSmith approximation for the Boltzmann equation (all calculations are done for Maxwell molecules) since it is believed that the MottSmith approximation yields better results for strong shocks. The comparison shows that GBE yield certain improvement of the NavierStokes results for moderate Mach numbers
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| 10. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Solutions of the linear Boltzmann equation and some Dirichlet series
- 2012
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Ingår i: Forum Mathematicum. - : Walter de Gruyter. - 1435-5337 .- 0933-7741. ; 24:2, s. 239-251
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a classof solutions of the Fourier transformed spatially homogeneous linear Boltzmannequation with a special Maxwell-type collision kernel. The result is based on anexplicit integral representationof solutions to the Cauchy problem for the Boltzmann equation. Possibleapplications to the theory of Dirichlet seriesare briefly discussed.
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